A heterogeneous nonlocal advection--diffusion system

TL;DR

This paper investigates the well-posedness of a nonlocal advection-diffusion system modeling heterogeneous populations, establishing local and global solutions under various kernel regularity conditions, and illustrating the effects numerically.

Contribution

It provides a comprehensive analysis of existence, uniqueness, and boundedness of solutions for a nonlocal system with heterogeneous kernels, including new results for irregular kernels.

Findings

Local well-posedness using semigroup theory

Global bounds for regular kernels via Nash inequality

Global boundedness with small initial data for irregular kernels

Abstract

We present a self-contained investigation on the local and global well-posedness for a system of nonlocal advection--diffusion equations for a heterogeneous population over $\mathbb{R}^d$, $d \in \mathbb{N}$. Each convolution kernel $K_{ij}$, which describes the nonlocal advection of species $i$ according to the distribution of species $j$, is assumed to have its own regularity $\nabla K_{ij} \in L^{q_{ij}}(\mathbb{R}^d),\, 1 < q_{ij} < \infty$. Local well-posedness of the mild solution and its regularity is obtained using semigroup theory and contraction mapping arguments. For families of kernels classified as regular, a global bound is established using a Nash-type inequality. For suitable irregular families of kernels, global boundedness is instead obtained via a smallness condition on the initial data. A one-dimensional numerical example is provided to illustrate the influence of kernel regularity on the solutions.